The connections on a PCB should be identical to its corresponding circuit diagram, but while the circuit diagram is arranged to be readable, the PCB layout is arranged to be functional, so there is rarely any visible correlation.

Using a via can make connection with ground plane.In case of surface-mount
component, there is separate via connected to the surface-mount pad.
In case of throughole component, the pin is soldered into the via.

If you put a big copper plane for ground on layer 2 and another one for power on 3 and assign them to the ground and power nets respectively. When you glood the copper, PADS will connect them as you want.